Table 3.10

Radii (nm) of the Inner and Outer Nanotubes for the Optimized C(5, 5)@BN(n, n) and C(9, 0)@BN(m, 0) [76]

n	BN(n, n)			C(5, 5)			R
	Rf	Rr	δR	Rf	Rr	δR
9	0.62142	0.63600	0.01458	0.34094	0.33368	−0.00726	0.30233
10	0.69003	0.68967	−0.00036	0.34094	0.34075	−0.00019	0.34892
11	0.75863	0.75624	−0.00239	0.34094	0.34175	0.00081	0.41449
12	0.82712	0.82543	−0.00169	0.34094	0.34143	0.00048	0.48400
13	0.89578	0.89484	−0.00094	0.34094	0.34121	0.00027	0.55362
14	0.96428	0.96382	−0.00047	0.34094	0.34107	0.00012	0.62275

m	BN(m, 0)			C(9, 5)			R
	Rf	Rr	δR	Rf	Rr	δR
15	0.59842	0.62725	0.02883	0.35409	0.33806	−0.01603	0.28919
16	0.63808	0.6513	0.01322	0.35409	0.34614	−0.00795	0.30515
17	0.67781	0.68094	0.00312	0.35409	0.35206	−0.00203	0.32888
18	0.71744	0.71566	−0.00179	0.35409	0.35451	0.00042	0.36115
19	0.75344	0.75066	−0.00278	0.35409	0.35567	0.00158	0.39499
20	0.79265	0.79034	−0.00231	0.35409	0.35496	0.00087	0.43537
21	0.83636	0.83453	−0.00184	0.35409	0.3547	0.00060	0.47983
22	0.87612	0.87475	−0.00136	0.35409	0.35452	0.00043	0.52023

When n=9, the radii of the outer BN(9, 9) for C(5, 5)@BN(9, 9) and of the free BN(9, 9) are 0.63600 and 0.62142 nm, respectively, the former being larger than the latter by 0.01458 nm. The radii of the inner C(5, 5) for C(5, 5)@BN(9, 9) and the free C(5, 5) are 0.33368 and 0.34094 nm, the former being less than the latter by 0.00726 nm. The space between the inner and outer nanotubes is 0.34892 nm. It is obvious that the radius of C(5, 5) decreases and that of BN(9, 9) increases when they are combined into C(5, 5)@BN(9, 9). This indicates the occurrence of greater repulsive interactions between the inner CNT and the outer BNNT. When n=10, the radii of the outer BN(10, 10) for C(5, 5)@BN(10, 10) and of the free BN(10, 10) are 0.68967 and 0.69003 nm, the former being only slightly smaller than the latter by 0.00036 nm. The radius of the inner C(5, 5) for C(5, 5)@BN(10, 10) is 0.34075 nm, which is also slightly less than that of the free C(5, 5) by 0.000194 nm. The space between the inner and outer nanotubes is 0.34892 nm. It is obvious that the radii of the inner and outer nanotubes are very close to those of the free structures. This indicates that the interaction between the inner CNT and the outer BNNT approaches zero. When n=11, the radii of the outer BN(11, 11) for C(5, 5)@BN(11, 11) and the free BN(11, 11) are 0.75624 and 0.75863 nm, the former being smaller than the latter by 0.00239 nm. The radius of the inner C(5, 5) for C(5, 5)@BN(10, 10) is 0.34175 nm, which is larger than that of the free C(5, 5) by 0.000805 nm. The space between the inner and outer nanotubes ($\mathrm{\Delta }R$) is 0.41449 nm. It is obvious that the radius of C(5, 5) increases and that of BN(11, 11) decreases when they are combined into C(5, 5)@BN(11, 11). This again indicates greater attraction interactions between the inner CNT and the outer BNNT. As the parameter n increases, this attraction interaction decreases gradually and tends to zero. Consequently, C(5, 5)@BN(10, 10) is a relatively stable armchair structure.

The situation for the zigzag C(9, 0)@BN(m, 0) (m=15–21) nanotubes (see Fig. 3.31) is similar to that for the armchair C(5, 5)@BN(n, n) already described. As parameter m increases, the interactions between the inner CNT and the outer BNNT switch from repulsion to attraction, although the transition is smoother. The corresponding transition chiral parameter m is about 17–18 and the space between the inner and outer nanotubes ($\mathrm{\Delta }R$) is 0.33–0.36 nm. Thus, the most stable zigzag structures are C(9, 0)@BN(17, 0) and C(9, 0)@BN(18, 0). Nanotubes are classified as zigzag, armchair, or chiral types according to their chiral angle. The chiral angle of zigzag nanotubes is 0 degree, whereas that of armchair nanotubes is 30 degrees. The chiral angle of chiral nanotubes is within 0 and 30 degrees. Comparing the foregoing armchair C(5, 5)@BN(n, n) with zigzag C(9, 0)@BN(m, 0) (see Figs. 3.30 and 3.31), it is found that the interaction between the inner and outer tube for armchair C(5, 5)@BN(10, 10) is much closer to zero than that for zigzag C(9, 0)@BN(17, 0) or C(9, 0)@ BN(18, 0). Therefore, the optimal interwall distances in C@BNNTs are related to their chiralities. The greater the chiral angle is, the smaller the interwall interaction of combination structure is. It is obvious that the armchair C@BNNTs achieve a more stable combination structure than the zigzag case.

In addition, the radii of the outer BN(10, 10) in optimal C(5, 5)@BN(10, 10) and the free BN(10, 10) are 0.68967 and 0.69003 nm; the former being slightly smaller than the latter by 0.00036 nm. Thus it is concluded that the BNNT in optimal C@BNNT is hardly influenced by the combination structure. Similar to the radii of BNNT, the structural parameter of CNTs in optimal C@BN is barely affected by the coating BNNTs. They display the same fundamental characteristics as CNTs in the free space. The BNNT around the CNTs effectively helps to protect them and makes them more stable, such as oxidation degradation of CNTs is reduced by coating with BNNTs, and the thermal stability of C@BNNTs is far superior to CNTs [83–86].

To explain the interactions between the inner CNT and the outer BNNT thoroughly, the radial distribution function (RDF) of the C(5, 5) and BN(n, n) nanotubes in the C(5, 5)@BN(n, n) (n=9, 10, and 11) structure are selected for analysis (see Fig. 3.32 and Table 3.11) [22]. In the RDF of nanotubes, the peak locations (abscissa values) correspond to the first neighbor, second neighbor, and other neighbor distances between the atoms of the hexagonal atomic ring on the tube wall. Fig. 3.32 (Table 3.12) shows first and second neighbor distances for the C(5, 5) in C(5, 5)@BN(10, 10) to be 0.141 and 0.243 nm and those for the BN(10, 10) in C(5, 5)@BN(10, 10) to be 0.145 and 0.249 nm, respectively. Compared with these data in C(5, 5)@BN(10, 10), Fig. 3.32A shows that first and second neighbor distances for the C(5, 5) in C(5, 5)@BN(9, 9) are reduced to 0.139 and 0.239 nm, and the other neighbor distances also decrease. However, Fig. 3.32B shows the opposite effect for the BN(9, 9). Its first and second neighbor distances increase to 0.147 and 0.253 nm, respectively. It is clear in this case that larger mutual interactions occur between the C(5, 5) and the BN(9, 9). Fig. 3.32A and B also reveal that the first neighbor distances of the C(5, 5) increase to 0.143 nm and that of BN(11, 11) are reduced to 0.143 nm in the optimized C(5, 5)@BN(11, 11), but the second and other neighbor distances only change slightly. This indicates a weak attractive interaction between the C(5, 5) and BN(11, 11). These results concur with those shown in Fig. 3.30.

3.4.2.3 Binding energy

The stability of C@BNNT is determined by the interaction force between the inner and outer nanotubes and the deformation caused by the force. The interaction energy E_i and its van der Waals energy E_iv between the inner and outer nanotubes and the binding energy E_b of each C@BNNT are calculated, as shown in Fig. 3.33. Tables 3.12 and 3.13 show the energy of C@BNNT and its inner CNT and outer BNNT. Their total potential energies are denoted by E_@, E_C, and E_BN, respectively, and their van der Waals energies are denoted by E_@v, E_Cv, and E_BNv, respectively. E_Cf and E_BNf represent the total potential energy of the inner CNT and the outer BNNT in their free state. E_C and E_BN are obtained by calculating the single point energy of the inner CNT and the outer BNNT in a C@BNNT. The interaction energy E_i (E_i=E_@–E_C–E_BN) between the inner and outer nanotubes is calculated. The binding energy E_b (E_b=E_@–E_Cf–E_BNf) of each C@BNNT is also calculated, in which the E_Cf of C(5, 5) and C(9, 0) is 6754.7044 and 5756.2738 KJ/mol, respectively. E_iv (E_iv=E_@v–E_Cv–E_BNv) denotes the van der Waals interaction between the inner and outer nanotubes, and is a component of the interaction energy E_i.

Fig. 3.33 shows that E_b is a positive value only when n=9, m=16. This indicates that the interaction force between the inner CNT and outer BNNT is one of repulsion. However, the interaction energy remains slightly negative. Only when m(n) is further reduced does a positive E_b occur. This is probably because the single point energy of the inner CNT and the outer BNNT in a C@BNNT is influenced by the H bond at either ends of the actual deformation, which means that the computed values of E_C and E_BN are slightly larger than their actual values. For a C@BNNT with $n\ge 10$, $m\ge 17$, E_b and E_i are always negative. This indicates that the interaction between the inner and outer nanotubes is one of attraction. Fig. 3.33 shows the curves for E_i and E_b with different n(m) for C(5, 5)@BN(n, n) and C(9, 0)@BN(m, 0). It can be clearly seen from the figure that the greatest attraction is displayed at n=10 for C(5, 5)@B(n, n) and at m=17–18 for C(9, 0)@BN(m, 0). At this point, the space between the tube walls for the armchair and zigzag C@BNNTs are about 0.35 and 0.33–0.36 nm, respectively. These values are just within the distance between the layers of 0.33–0.36 nm for graphite and h-BN. When $n\ge 11$ and $m\ge 19$, the attraction between the inner CNT and the outer BNNT begins to decline. As E_b is the energy change when the free CNT and free BNNT are bound together as a C@BNNT system, n=9 and m=15 and 16 result in endothermic processes, and the rest are exothermic processes. The heat of the exothermic process is highest when n=10 and m=17–18. This indicates that the systems with n=10 and m=17–18 are most stable. The difference between E_i and E_b reflects the energy loss caused by the deformation of the CNT and BNNT when the free CNT and free BNNT are bound together as a C@BNNT. Fig. 3.33 shows that the maximal energy loss takes place in the C@BNNT, where n=9 and m=15 and 16. This is because there is greater repulsion between the inner and outer nanotubes, which leads to greater deformation and results in a larger energy loss. It should be noted that when n=8 (not shown in the Fig. 3.33), the C(5, 5)@BN(n, n) structure cannot be maintained because of the powerful repulsion between the inner and outer tubes. The optimal structure occurs with a separation along the axial direction (see Fig. 3.29). When $n\ge 11$or $m\ge 19$, the distances between the inner and outer nanotubes increase, and the deformation and energy loss decrease rapidly. In the formation of the C(5, 5)@BN(n, n) and C(9, 0)@BN(m, 0) systems, the energy change derives mainly from the interaction force between molecules, and the effect of deformation on the energy change is slight. A comparison of E_i and E_iv in Fig. 3.33 shows that the difference is very small. It can thus be concluded that the interaction between the inner CNT and outer BNNT is a typical van der Waals interaction between molecules.

3.4.2.4 Electronic structure and bonding model

To further understand the interaction between the CNT and BNNT in a C@BNNT, the deformation electron density and its isoline structure are calculated. As the effects of different values of the parameter n(m) on the electronic distribution of C(5, 5)@BN(n, n) and C(9, 0)@BN(m, 0) follow a similar law, only C(5, 5)@BN(n, n) is used in the analysis. Fig. 3.34 shows the deformation electron density and its section isoline as calculated by DFT by simulating a C(5, 5)@BN(n, n) model for n=9, 10, and 11. The electron density is the same as the online equivalent, and gradually becomes smaller on the vertical isolines from the inside outward. The isoline intervals reflect the local electronic conditions, in that the greater the interval, the weaker the localization and the more free the electrons, and vice versa.

As shown in Fig. 3.34, in the outer tube BN(n, n)NT, the outer-layer electrons are obviously partial to the N atom in the B–N bond. However, in the inner CNT, the electrons are more concentrated at the center of the C–C bond. The outer-layer electrons are the main cause of the covalent bond. If the outer-layer electrons spend most of their time within the nuclei of the two atoms, then the covalent bond is placed in the bonding orbital, in which case the distribution shape of the electron density is small at either end and large in the middle. In contrast, if the electrons spend most of their time outside the nuclei of the two atoms, then the covalent bond is placed in the antibonding orbital, in which case the distribution shape of the electron density is large at either end and small in the middle. This is because in the bonding orbital there is an increase in electron density between the nuclei, whereas in the antibonding orbital there is a decrease in electron density. Placing an electron in the bonding orbital stabilizes the molecule because it remains in between the two nuclei. Conversely, placing electrons in the antibonding orbital cause a decrease in stability [87]. The deformation electron density plot in Fig. 3.34 shows that C(5, 5)@BN(10, 10)NT has a good bonding electronic distribution that is small at either end and large in the middle. The electrons are uniformly distributed along the atoms in rings. The shape when n=11 is similar to that when n=10, but is different when n=9. This is mainly because a part of the bond of the inner CNT shows an antibonding effect. The isoline structure in Fig. 3.34 shows some regular change as the parameter n increases. For example, when n=9, because of the repulsive interactions between the inner CNT and the outer BNNT, some of the outer electrons tend toward the inside. Consequently, the isoline interval is smaller and a partial antibonding effect may occur. When n=10, the repulsive interactions disappear, the outer layer regains its electron density, the isoline interval is medium-sized, and it is difficult for the antibonding effect to occur. The electron distribution also has a good bonding characterization. Thus, the structural stability of C(5, 5)@BN(10, 10) is superior to that of C(5, 5)@BN(9, 9). When n=11, due to the attraction between the CNT and the BNNT, the electronic localization of the inner layer may be weakened, some s electrons become p electrons due to excitation, and partial electronic delocalization of the outer layer occurs, which results in an increase in electron mobility. Consequently, the outside electron density for n=11 decreases and the isoline interval for the inner layer become somewhat sparse. Of course, as this effect is small, the electron density distribution is still similar to that when n=10. In general, the numbers of nonlocalized outer-layer electrons, which are related to the conductivity, increase, and the number of bonding electrons, which maintain the stability of the structure, naturally decrease. Thus, the structural stability of C(5, 5)@BN(10, 10) is superior to that of C(5, 5)@BN(11, 11).

Fig. 3.34 clearly shows that bonding electronic distributions are only formed between B and N or C and C, and no bond is formed between B and C or N and C. The distribution state remains basically unchanged despite changes in parameter n. This indicates that there is no bond mix between the CNT and the BNNT in a C@BNNT, and the interactions between the CNT and BNNT are only van der Waals forces between the molecules. This result concurs with the conclusion drawn from the energy analysis.

3.5.3.1 Twisted CNT bundles under axial compression

When a twisted CNT bundle is under axial compression, the six surrounding SWCNTs will expand and contract radially around the midlength along the center axis of the CNT bundle. To quantify the radial expansions and contractions and the morphological changes of the CNT bundle, the total intertube van der Waals force acting on the core SWCNT ($F_{vdw}^{core}$) of each CNT bundle twisted at various angles (i.e., ${\theta }_t$=0, 30, 60, and 90 degrees) and its deformation $\mathrm{\Delta }{l}_t$ (i.e., defined as $\mathrm{\Delta }{l}_t$=(l_t−l)/l_t, where l is the current length of the CNT bundle), were computed and plotted in Fig. 3.49. The graph shows that $F_{vdw}^{core}$ undergoes a cyclic pattern during compression. When the six surrounding SWCNTs expand radially, $F_{vdw}^{core}$ becomes weaker as shown by the “valleys” of the plots in Fig. 3.49. However, when the six surrounding SWCNTs contract radially, $F_{vdw}^{core}$ grows stronger, as shown by the “peaks” of the plots in Fig. 3.49.

A twisted CNT bundle is expected to have a lower critical buckling load P_cr and critical strain ${\epsilon }_{cr}$ than a nontwisted CNT bundle due to the curved geometries of the six surrounding SWCNTs that are formed by the twisting. Fig. 3.50 depicts the plot of the critical buckling load P_cr against the angle of twist ${\theta }_t$ in the CNT bundle. As the twist angle ${\theta }_t$ increases from 0 to 60 degrees, the critical buckling load P_cr decreases by about 5% from 325 to 308 nN. A further increase in the twist angle ${\theta }_t$ from 60 to 120 degrees, however, results in a larger drop of about 42% in the critical buckling load P_cr from 325 to 188 nN. This is because as the twist angle ${\theta }_t$ grows, the compressive force at each end of the surrounding SWNCT becomes more misaligned, which causes the surrounding SWNCT to buckle more easily and at a lower load P_cr.

Additionally, it is also interesting to note that the six surrounding SWCNTs will buckle earlier than the core SWCNT due to their curved geometries. Fig. 3.51 shows the morphological changes of a 60 degree-twisted CNT bundle at different deformations $\mathrm{\Delta }{l}_t$. In the initial stage when $\mathrm{\Delta }{l}_t$=0, the whole CNT bundle is relaxed, and the helical angle ${\theta }_{he}$ that is formed due to the twisting of the six surrounding SWCNTs is about 77 degrees. Upon further compression, up to when the deformation $\mathrm{\Delta }{l}_t$=0.048 (see Fig. 3.51B), the six surrounding SWCNTs buckle outwards from the center axis fractionally earlier and form a “bird-cage” pattern, while the core SWCNT remains unbuckled. The fully buckled CNT at $\mathrm{\Delta }{l}_t$=0.06 is shown in Fig. 3.51C.

To gain further insight into the axial load P that each SWCNT experiences in the twisted CNT bundle during compressive loading, the axial force F of each individual SWCNT in the CNT bundle is plotted against its deformation $\mathrm{\Delta }{l}_t$ as in Fig. 3.52. As the axial force F that acts on the six surrounding SWCNTs is the same, therefore only the axial force F that acts on one surrounding SWCNT and the core SWCNT is plotted. It is evident from the plot that the surrounding SWCNT buckles slightly earlier than the core SWCNT and at a lower critical buckling load Pcr. This is because since the surrounding SWCNT is geometrically twisted, therefore it is more susceptible to buckling than the core SWCNT.

3.5.3.2 Twisted CNT bundles under axial tension

In contrast to wire ropes, which provide better structural reliability and are mainly used in applications in which the tensile load is dominant, twisted CNT bundles tend to underperform when compared with nontwisted CNT bundles subjected to an axial tensile load. As a result of the twisting, the six surrounding SWCNTs are brought closer to each other under a helical angle, and junctions are formed between the neighboring SWCNTs. During axial tensile loading, the six surrounding SWCNTs were pulled even closer to the core SWCNT, which induced a reduction in the intertube distance d_in between individual SWCNTs. The plot in Fig. 3.53 shows the relationship between the intertube distance d_in and strain $\epsilon$, which is computed as $\epsilon$=(l−l_t)/l_t, where l and l_t are the current and initial length of the CNT bundle, respectively. From the plot, it is evident that the intertube distance d_in decreases as the strain $\epsilon$ increases for CNT bundles with a large twist angle ${\theta }_t$. This is because when the six surrounding SWCNTs are twisted at a larger angle ${\theta }_t$, they are more closely intertwined and are geometrically curved more prominently. Consequently when a tensile load is applied to both ends of the CNT bundles, the curved geometry of each of the six individual surrounding SWCNTs tends to straighten out, which leads to a reduction in the intertube distance d_in.

This phenomenon can also be confirmed by the plot of $F_{vdw}^{core}$ (total intertube van der Waals force $F_{vdw}^{core}$ acting on the core SWCNT) at different strains $\epsilon$, as shown in Fig. 3.54. When a tensile load is applied to the CNT bundle, $F_{vdw}^{core}$ increases until it reaches a maximum value. This indicates a decrease in the intertube distance d_in between the surrounding six SWCNTs and the core SWCNT. $F_{vdw}^{core}$ then slowly decreases, which is not a result of an increase in the intertube distance between the six surrounding SWCNTs and the core SWCNT, but is an indication of an increase in the repulsive energy, which is explained further below.

In contrast to the theory of wire ropes in a continuum domain, twisted CNT bundles do not exhibit better tensile properties, as shown in Fig. 3.55. When the twisting angle ${\theta }_t$ is greater than 75 degrees, the failure load P_fl decreases sharply. An investigation into the cause of this was carried out, and it was found that when a CNT bundle with a twisting angle ${\theta }_t$ that is greater than 75 degrees is subjected to axial tension, the intertube distance d_in between the seven SWCNTs decreases to less than 0.2 nm. This results in an increase in the repulsive energy between the seven SWCNTs since the contact distance is less than 0.34 nm. Consequently, these CNT bundles tend to fail at the junctions at lower loads and strains. This is in qualitative agreement with previous work [89–92], in which it was found that the dangling C–C bonds that were formed between SWCNTs and which were brought close together were not strong enough to withstand bending and axial forces.

The total intertube van der Waals force $F_{vdw}^{total}$ and intertube repulsive force $F_{pre}^{total}$ around the midlength of the CNTs that has been twisted at 0 and 90 degrees were extracted from the MD simulations and are depicted in Figs. 3.56 and 3.57, respectively. The midlength of the CNT bundle was chosen for the depictions, as this is where pairs of the twisted surrounding SWCNTs meet each other at a helical angle and form a junction where they overlap. As the strain $\epsilon$ becomes larger, $F_{vdw}^{total}$ in the 90 degree-twisted CNT bundle increases due to the decreasing of intertube distance d_in (see Fig. 3.53). However, $F_{vdw}^{total}$ in the nontwisted CNT bundle weakens steadily as the strain $\epsilon$ increases due to the elongation of the CNT bundle during tension, as there is no change in the intertube distance d_in.

Throughout the whole tensile process, the nontwisted CNT bundle has no $F_{pre}^{total}$ (intertube repulsive force) due to the large intertube spacing d_in, as shown in Fig. 3.57. Likewise, there is no $F_{pre}^{total}$ present in the 90 degree-twisted CNT bundle that is twisted at 90 degrees up to $\epsilon$=0.185. This is because prior to $\epsilon$=0.185, the intertube distance d_in is greater than 0.2 nm, and, according to Eq. (3.15), the repulsive force F_rep is only computed when the distance between two atoms is less than 0.2 nm. However, after $\epsilon$>0.185, the intertube distance d_in for the 90 degree-twisted CNT bundle reduces to less than 0.2 nm (see Fig. 3.53), which results in a sudden increase in $F_{pre}^{total}$ and a decrease in $F_{vdw}^{total}$ (total intertube van der Waals force), as shown in Fig. 3.56. It is worth noting that $F_{vdw}^{total}$ and $F_{pre}^{total}$ act radially on neighboring SWCNTs. As a result of the sudden surge in $F_{pre}^{total}$, the 90 degree-twisted CNT bundle is weakened severely, and therefore only a low tensile load is needed to damage it. The damage often occurs at the junction areas (see Fig. 3.58).

As a consequence of the sudden increase in $F_{pre}^{total}$ at the junction areas at which the individual SWCNTs overlap, CNT bundles that are twisted at an angle ${\theta }_t$>75 degrees will collapse at lower critical strains ${\epsilon }_{\mathrm{cr}}$, as shown in Fig. 3.59. The plot shows that the critical strains remain at around ${\epsilon }_{cr}\approx$ 0.35 up to a twisting angle ${\theta }_t$ of 75 degrees, and decrease significantly thereafter to ${\epsilon }_{\mathrm{cr}}\approx$ 0.18.

Fig. 3.60 shows the cross-sectional view of two CNT bundles with a twisting angle of 0 and 120 degrees, respectively, at different strains. It is evident from this figure that the 120 degree-twisted CNT bundle suffers from a large amount of distortion of the individual SWCNTs compared with the nontwisted CNT bundle. The SWCNTs in the 120 degree-twisted CNT bundle lose their cylindrical shapes at high strains, which results in a dislocation of the hexagonal arrangement of the C atoms. The distortion of individual SWCNTs in the CNT bundle is manifested by the compressive van der Waals force F_vdW and the repulsive force F_rep between the seven SWCNTs.